I'll supply an answer to my own question, but I'm genuinely interested to see if others have better or different information than what I've been able to glean by semi-random searches through the voluminous literature.
The general impression I get is that cosmic censorship (CC) looks dead. The situation seems to be a little bit like the one in artificial intelligence, where people kept redefining AI over the years so that once a problem was solved, it was no longer considered to be "real" AI. With the passage of 40 years, more and more counterexamples have turned up, and in order to keep CC on life support, it has been necessary to add stricter and stricter assumptions.
In general, a meaningful definition of what it means to violate
weak cosmic censorship should probably include something
like the following ingredients.
- The initial conditions do not make available an infinite amount of energy within
a finite region.
- The initial conditions do not contain singularities.
- Incomplete lightlike geodesics can arrive at a distant observer.
- Such a violation still occurs if we impose small perturbations on the initial data.
- The forms of matter are physically realistic.
If we don't impose something like condition 1, then we can set up initial conditions that are of no interest because they are unrealistic. For this reason, one usually studies spacetimes that are asymptotically flat. It may also be necessary to impose a requirement that the matter fields fall off at some rate as we go to infinity. Condition 2 expresses the idea that any singularities that occur should be new ones formed by gravitational collapse. Usually one also assumes that no trapped surface has formed initially. The censorship violation is expressed by condition 3. If 4 is omitted, then clear counterexamples to censorship are known. However, it is not known whether there is an appropriately rigorous way to define ``small perturbations'' here. (In technical terms, we do not have any topology or measure defined on the set of all possible initial conditions. In actual work to date, people have selected some set of possible initial conditions, described by some small number of adjustable parameters, and have then tried to test condition 4 using a seemingly natural topology and measure defined on the space of those parameters.) Realistic matter fields, 5, are expected, for example, to satisfy certain energy conditions.
Because weak cosmic censorship seems to be violated if described by these five conditions (see, e.g., Joshi 2014), people have started looking for additional conditions that could salvage the conjecture.
Wald (1997) suggests adding a requirement that the types of matter be further restricted to ones having the property that if the metric is fixed, rather than dynamical as in general relativity, then no singularities occur. This seems to me to be much too strong a condition, since it rules out some forms of matter that may actually exist in astrophysical collapse, and in any case there are indications that it is not sufficient. (There are some remarks on this in the video seminar of Duffy 2011.)
Another proposal is along the following lines. When a naked singularity occurs, then we have a region of spacetime for which the singularity is inside the past lightcone. The lightlike surface constituting the boundary of this region is called a Cauchy horizon. An observer who passes beyond the Cauchy horizon can observe arbitrary information, i.e., phenomena not predicted by any laws of physics, and infinite fluxes of energy. Penrose has, however, pointed out that in certain illustrative cases, there is a tendency for energy from the entire spacetime prior to the singularity to be focused onto the Cauchy horizon. The result could then be that such an observer is destroyed when passing through the Cauchy horizon. In other words, the Cauchy horizon actually turns into a singularity. People have put a lot of effort into checking that this mechanism preserves CC in certain analytically tractable cases (Duffy 2011). The mechanism appears to fail, however, for a spacetime with a positive cosmological constant, which is what we actually have in our universe.
It looks to me like all the defenses are crumbling. There has even been some discussion of the possibility of actually going out and observing naked singularities (Joshi 2013, Kong 2013, Ortiz 2014).
Some fairly recent monographs and pedagogical or review papers on this topic are Harada 2001, Christodoulou 2008, Klainerman 2008, Joshi 2011, Isenberg 2015, and Baez 2016. Some of these are much more specialized than their titles would suggest. I found Joshi 2011 useful, but its treatment of many topics is mathematically superficial, and you need to read it while taking into account that Joshi is a hard-line anti-CC hawk.
Baez, "Struggles with the Continuum," 2016, https://arxiv.org/abs/1609.01421
Christodoulou, "The formation of black holes in general relativity," 2008, https://arxiv.org/abs/0805.3880
Duffy and Nolan, 2011, https://arxiv.org/abs/1108.1103 . There is also a video of Nolan giving a seminar on this work at http://pirsa.org/11040078 .
Harada, "Physical Processes in Naked Singularity Formation," 2001, https://arxiv.org/abs/gr-qc/0204008
James Isenberg, "On Strong Cosmic Censorship," 2015, https://arxiv.org/abs/1505.06390
Joshi and Malafarina, "Recent developments in gravitational collapse and spacetime singularities," 2011, https://arxiv.org/abs/1201.3660
Joshi et al., "Distinguishing black holes from naked singularities through their accretion disk properties," 2013, http://arxiv.org/abs/1304.7331
Joshi and Malafarina, "All black holes in Lemaitre-Tolman-Bondi inhomogeneous dust collapse," 2014, https://arxiv.org/abs/1405.1146
Klainerman, "Cosmic censorship and other great mathematical challenges of general relativity," https://web.math.princeton.edu/~seri/homepage/papers/BE2008.pdf
Kong, Malafarina, and Bambi, "Can we observationally test the weak cosmic censorship conjecture?," 2013, https://arxiv.org/abs/1310.8376
Ortiz, "Observational distinction between black holes and naked singularities: the role of the redshift function," 2013, http://arxiv.org/abs/1401.4227
Penrose, "The Question of Cosmic Censorship," J. Astrophys. Astr. (1999) 20, 233–248; http://www.ias.ac.in/jarch/jaa/20/233-248.pdf
Wald, "Gravitational Collapse and Cosmic Censorship," 1997, http://arxiv.org/abs/gr-qc/9710068